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Simplifying 9.1 = -0.01x2 + 0.25x + 8.10 Reorder the terms: 9.1 = 8.10 + 0.25x + -0.01x2 Solving 9.1 = 8.10 + 0.25x + -0.01x2 Solving for variable 'x'. Combine like terms: 9.1 + -8.10 = 1 1 + -0.25x + 0.01x2 = 8.10 + 0.25x + -0.01x2 + -8.10 + -0.25x + 0.01x2 Reorder the terms: 1 + -0.25x + 0.01x2 = 8.10 + -8.10 + 0.25x + -0.25x + -0.01x2 + 0.01x2 Combine like terms: 8.10 + -8.10 = 0.00 1 + -0.25x + 0.01x2 = 0.00 + 0.25x + -0.25x + -0.01x2 + 0.01x2 1 + -0.25x + 0.01x2 = 0.25x + -0.25x + -0.01x2 + 0.01x2 Combine like terms: 0.25x + -0.25x = 0.00 1 + -0.25x + 0.01x2 = 0.00 + -0.01x2 + 0.01x2 1 + -0.25x + 0.01x2 = -0.01x2 + 0.01x2 Combine like terms: -0.01x2 + 0.01x2 = 0.00 1 + -0.25x + 0.01x2 = 0.00 Begin completing the square. Divide all terms by 0.01 the coefficient of the squared term: Divide each side by '0.01'. 100 + -25x + x2 = 0 Move the constant term to the right: Add '-100' to each side of the equation. 100 + -25x + -100 + x2 = 0 + -100 Reorder the terms: 100 + -100 + -25x + x2 = 0 + -100 Combine like terms: 100 + -100 = 0 0 + -25x + x2 = 0 + -100 -25x + x2 = 0 + -100 Combine like terms: 0 + -100 = -100 -25x + x2 = -100 The x term is -25x. Take half its coefficient (-12.5). Square it (156.25) and add it to both sides. Add '156.25' to each side of the equation. -25x + 156.25 + x2 = -100 + 156.25 Reorder the terms: 156.25 + -25x + x2 = -100 + 156.25 Combine like terms: -100 + 156.25 = 56.25 156.25 + -25x + x2 = 56.25 Factor a perfect square on the left side: (x + -12.5)(x + -12.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (x + -12.5) equal to 7.5 and -7.5.Subproblem 1
x + -12.5 = 7.5 Simplifying x + -12.5 = 7.5 Reorder the terms: -12.5 + x = 7.5 Solving -12.5 + x = 7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12.5' to each side of the equation. -12.5 + 12.5 + x = 7.5 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + x = 7.5 + 12.5 x = 7.5 + 12.5 Combine like terms: 7.5 + 12.5 = 20 x = 20 Simplifying x = 20Subproblem 2
x + -12.5 = -7.5 Simplifying x + -12.5 = -7.5 Reorder the terms: -12.5 + x = -7.5 Solving -12.5 + x = -7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '12.5' to each side of the equation. -12.5 + 12.5 + x = -7.5 + 12.5 Combine like terms: -12.5 + 12.5 = 0.0 0.0 + x = -7.5 + 12.5 x = -7.5 + 12.5 Combine like terms: -7.5 + 12.5 = 5 x = 5 Simplifying x = 5Solution
The solution to the problem is based on the solutions from the subproblems. x = {20, 5}
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